How do you use the mean value theorem to explain why (-1) is the only zero of the function f(x)=x³+3x²+9x+7?

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Answer 1 · 2017-02-23T23:24:08

Please see below.

If there were another zero, say at #b#,

then we could apply the mean value theorem on the interval #[b,-1]# (if #b < -1#) or on #[-1,b]# (if #b > 1#)

to conclude the=at there is a #c# in the corresponding open interval at which #f'(c) = 0#

But #f'(x) = 3x^2+6x+9# has no real zeros.

Therefore, there cannot be two zeros of #f#.